Does not compute. Your equipment melds into your form, effectively depriving you of any components or foci. While elementals can speak, the jury is still out whether or not they can gesture well enough to employ somatic components of a spell.

If you have Eschew Materials or just have a friend hold your component pouch till after you transform (god the polymorph rules are stupid), you can cast. Elementals can speak and have or can assume ahumanoid-shaped form. For fire elemental specifically, you'd need some fire resistance on the pouch to keep it on your person, though. :)

I have no idea on hit points, if they're unchanged or the con score boost changes them. In 3E, they remained the same, PF has no notes one way or the other, so I guess they would increase until the spell ends.

Look up the spell it spells put what you gain you may have to look back at the lower level versions of the spell as each one grants the lower version and new stuff. As for spell casting as far as casting it's up to the GM a druid needs a feat to cast in wild shape so normally no but elementals can speak so any verbal only spells would work so just ask nicely. Hp goes up and down with temp con changes that is how it works it's spelled out very clearly look at anything that changes con, belts, rage, etc.

Kept. Some of your skills might be unusable, due to the situation, though. (Swim, anyone?)

Actually you can swim just as normal while under the effect of the spell, because you are not really a Fire Elemental, which cannot touch water. You are emulating the Fire Elemental form and abilities and not becoming a true elemental.

The spell doesn't even gain the Fire descriptor so you can freely cast it while underwater (or ativate the spell-like ability in the case of the Oracle).

...But a true fire elemental might (an hypothesis) not enter in water because it is an elemental (and it has the subtype) and elementals are entirely made of their element (in this case fire) that would extinguish underwater. The spell doesn't give the caster a creature subtype.